How do you know if it's psychosomatic or not?True! I was referring to the psychosomatic 'seven holy cows' (gastric ulcera, colitis ulcerosa, neurodermatitis, asthma, hypertension, rheumatoid arthritis, thyreoid dysfunction) which is of course a historic classification. All those are under influence of you current state of mind and have clear somatic correlates (e.g. TPO-Antibodies for Hashimoto) but often we do not know where they are coming from. Of course there is no measure regarding how dependent on your state of mind a disease needs to be in order to be classified as psychosomatic.

How repeatable are cognitive science findings within the same individual?I think that this question is very broad and could be improved if you specified which area you are interested in. As Steven pointed out running the same experiment twice in human subjects is done when effects of learning are investigated. If no learning effects are expected and performance is hypothesized to be stable, researchers often investigate 'retest-reliability'. So if you google/pubmed the area you are intetested in (e.g. memory) plus 'learning' or 'retest-reliability' you will find lots of studies.